Cycles in Graphs by B. R. Alspach

By B. R. Alspach

This quantity bargains with a number of difficulties related to cycles in graphs and circuits in digraphs. prime researchers during this quarter current right here three survey papers and forty two papers containing new effects. there's additionally a suite of unsolved difficulties.

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P a i r f o r cycles {C,D} l i n k i n g e i t h e r switchable v i a * P,Q S,T o r joinable v i a . Then {C,D} i s The h y p o t h e s e s o f e i t h e r Lemma 8 o r Lemma 9 are s a t i s f i e d . 0 Proof. DEFINITION. Let be a cycle i n a c o l l e c t i o n C {P,Q} be d i s j o i n t c h o r d a l p a t h s o f P U Q U S U T = C’ such t h a t replace G . P a removable {S,T) in t h e cubic multigraph P in C X C . L e t {S,T} is a cycle. w i t h t h e set of c y c l e s o f c y c l e s and l e t X b e two p a t h s l y i n g i n To switch via C * S,T C is t o P,Q 2 = C A C’.

T h i s completes t h e s u b c a s e t h a t orm = 1. i s j u s t the vertex to 29 0 uo am = -1. Now c o n s i d e r t h e s u b c a s e t h a t Si = {O} whenever Si # 0 and 1 5 i 5 1-1 Notice t h a t t h i s implies t h a t . I n o r d e r t o complete t h e proof of t h i s s u b c a s e , w e must i n t r o d u c e some n o t a t i o n and a lemma. , ~ , v v~ } n-1 {uiui+l: G(n,2) G(n,k) has vertex-set and edge-set 0 5 i 5 n-1) U { v ~ v ~ +0 ~5 : i 5 n-1). h a s a H a m i l t o n c y c l e i f and o n l y i f 0 5 i 5 n-11 U {uivi: n N e i l Robertson [61 proved t h a t 5 (mod 6 ) and Kozo Bannai [ 4 ] proved t h a t G(n,k) h a s a Hamilton c y c l e whenever n k and a r e r e l a t i v e l y prime e x c e p t i n t h e c a s e s t h a t n-1 n+l n S 5 (mod 6 ) and k = 2,-2, 2 LEMMA.

Uo u1 Type D i n s e r t i o n uO u2 - becomes - u2 11. The u1 - - 1 A w 7 v A v Type E i n s e r t i o n Figure 1 We now f i n d a p a t h from case rn G(11,2) = is shown i n F i g u r e 2. g i v e s a Hamilton p a t h from Let n of t y p e uo v to 1 is a l r e a d y t a k e n care o f . b e t h e smallest C starting at n E 5 (mod 6 ) where n 2 n u A sequence of type to *lo. E n and to uo ? v 7 in i n s e r t i o n s ( s e e F i g u r e 1) v ~ - f o~r e a c h n 2 1 7 , n Z 5 (mod 6 ) .

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